Tilting Subcategories And Grothendieck Groups Of Extriangulated Categories

The thesis includes two parts.In the first part,we study tilting subcategories in an extriangulated category（called E-category for short）;in the second part we consider the Grothendieck group of an E-category.In the first part,we give the definition of an n-tilting subcategory in an E-category;firstly,we prove that the Bazzoni characterization of an n-tilting subcategory also holds in E-categories;then we prove the Auslander-Reiten correspondence of n-tilting subcat-egories in an E-category,that is,there is a one-to-one correspondence between n-tilting subcategories and coresolving covariantly finite subcatgories which are closed under di-rect summands and satisfyˇX_n=C.As main applications of the above results:on the one hand,we suppose that A is an abelian category with a complete hereditary cotor-sion triple,and we reprove Wei et al’s^[26]results of Bazzoni characterization and the Auslander-Reiten correspondence of tilting subcategories in the relative homological cat-egory of A;on the other hand,we consider the triangulated category with a proper class of triangles introduced by Beligiannis^[16],and prove that C together with the proper class of triangles forms an E-category E,if in addition,E has enough projectives and injec-tives,we can get the Bazzoni characterization and the Auslander-Reiten correspondence of tilting subcategories in the relative homology sense of a triangulated category.In the second part,when C is a locally finite E-category,we prove that C has Auslander-Reiten E-triangles and the relation group of its Grothendieck group K₀（C）is generated by Auslander-Reiten E-triangles.Then we consider the converse of the above result:we firstly show that when restricting to a triangulated category C with a cluster tilting subcategory,the relation group of K₀（C）is generated by Auslander-Reiten E-triangles if and only if C is locally finite;then we consider the relative Grothendieck group K₀（C,T）of a triangulated category C with a cluster tilting subcategory T,we prove that the relation group of K₀（C,T）is generated by some special Auslander-Reiten E-triangles if and only if C is locally finite.At the last of this part,we consider the E-category C with a generating category G,and prove that we can use subgroups of K₀（C）which contain the image of G to classify dense G-（re）solving subcategories in C.